Global well-posedness and time-decay of solutions for the compressible Hall-magnetohydrodynamic system in the critical Besov framework

We consider the global existence of solution for the initial value problem for the compressible Hall-magnetohydrodynamic system in the whole space R³. The system consists of a hyperbolic-parabolic system of partial differential equations of the conservation laws type with non-symmetric diffusion. We show the existence of solution as a perturbation from a constant equilibrium state (ρ¯,0,B¯), where ρ¯>0 is a constant density, 0∈R³ is the zero velocity and B¯∈R³ is a constant magnetic field. The time-decay of the solution in the Besov spaces is also established. Our results show the pointwise estimate of the solution in the Fourier space for the linearized Hall-MHD system that related to the result obtained by Umeda–Kawashima–Shizuta [33] for a general class of linear symmetric hyperbolic-parabolic systems with symmetric diffusion. We utilize a systematic use of the product estimates in the Chemin–Lerner spaces and apply the energy method due to Matsumura–Nishida [27].